The Bott periodicity theorem states that $\pi_k(O(\infty))=\pi_{k+8}(O(\infty))$ and similarly for other classical Lie groups.
But his groups are defined as an inductive limit. But for e. g. $GL(n,F)$ the inductive limit is not the same as the group of all invertible transformations of the Hilbert space.

(1) What can we say about the group of all transformations of $F^\infty$ in the context of Bott periodicty?

(2) What are the Lie algebras of the two aforementioned groups? Are they somehow related?

(3) Is it possible to represent some of these groups as an "infinite Dynkin diagram"? (For example, I would imagine $sl(\infty)=A_\infty$ being represented by an infinite row of connected circles.

The Bott periodicity theorem states that $\pi_k(O(\infty))=\pi_{k+8}(O(\infty))$ and similarly for other classical Lie groups.
But his groups are defined as an inductive limit. For e.g. $GL(n,F)$, for $F=\mathbb{R}$ or $\mathbb{C}$, the inductive limit is not a priori the same as the group of all invertible transformations of $F^\infty$. Besides, it's certainly not isomorphic to the group of all invertible transformations of the (separable infinite dimensional) Hilbert space.

(1) What can we say about the group of all transformations of $F^\infty$ in the context of Bott periodicty?

(2) What are the Lie algebras of the aforementioned groups? Are they somehow related?

(3) Is it possible to represent some of these groups as an "infinite Dynkin diagram"? (For example, I would imagine $sl(\infty)=A_\infty$ being represented by an infinite row of connected circles).